ON P-VALENT STRONGLY STARLIKE AND STRONGLY CONVEX FUNCTIONS
Ali muhammad
Abstract. In this paper, we define some new classes S∗(a, c, λ, p, α, β) and
C(a, c, λ, p, α, β) of strongly starlike and strongly convex functions or order αand type β by using Cho- Kown -Srivastava integral operators. We also derive some interesting properties, such as inclusion relationships of these classes.
2000Mathematics Subject Classification: 30C45, 30C50.
Keywords: Analytic functions; Strongly srarlike functions; Strongly convex func-tions; Cho Kown Srivastava integral operators.
1. Introduction
LetA(p) denote the class of functionsf(z) normalized by
zp+ ∞
X
k=1
ap+kzp+k, p∈N={1,2,3, ...}. (1.1)
which are analytic and p-valent in the unit diskE ={z:z∈Cand |z|<1}.
A function f(z) ∈A(p) is said to be in the class S∗(p, β) of p-valently starlike function of order β inE if it satisfies the following inequality:
Re
zf′(z)
f(z)
> β, z∈E, 0≤β < p. (1.2) The class S∗(p, β) was introduced and studied by Goodman [4], see also [11]. For recent investigation and more detail the interested reader are refers to the work by [12, 13]. Also, we note thatS∗(p,0) =S∗
p,whereSp∗is the class of p-valently starlike
A function f(z) ∈ A(p) is said to be in the class C(p, β) of p-valently convex function of order β inE if it satisfies the following inequality:
Re
1 +zf ′′(z)
f′(z)
> β, z∈E, 0≤β < p. (1.3) The class C(p, β) was introduced and studied by Goodman [4], see also [10]. Also we note that C(p,0) = Cp, where Cp is the class of p-valently convex functions in
E.
A function f(z) ∈A(p) is said to be strongly starlike of order α and type β in
E if it satisfies the following inequality:
arg
zf′(z)
f(z) −β
< απ
2 , z ∈E, 0≤α < pand 0≤β < p. (1.4) We denote this class by S∗(p, α, β).A function f(z) ∈ A(p) is said to be strongly convex of order α and type β inE if it satisfies the following inequality:
arg
1 +zf ′′(z)
f′(z) −β
< απ
2 , z ∈E, 0≤α < pand 0≤β < p. (1.5) We denote this class by C(p, α, β).
It is obvious that f(z) ∈ A(p) belongs to C(p, α, β) if and only if zff(′z(z)) ∈
S∗(p, α, β).Also, we note thatS∗(p, α, β) =S∗(p, β) and C(p, α, β) =C(p, β). In our present investigation we shall make use of the Gauss hypergeometric functions defined by
2F1(a, b;c;z) = ∞
X
k=0
(a)k(b)kzk
(c)k (1)k
, (1.6)
wherea, b, c∈C,c /∈Z−
0 ={0,−1,−2, ...}and (k)ndenote the Pochhammer symbol
(or the shifted factorial) given, in terms of the Gamma function Γ, by (k)n=
Γ(k+n) Γ (k) =
k(k+ 1)...(k+n−1) n∈N,
1 n= 0.
We note that the series defined by (1.6) converges absolutely for z ∈E and hence represents an analytic function in the open unit disk E, see [14].
We define a functionφp(a, c;z) by
φp(a, c;z) =zp+ ∞
X
k=1
(a)k zp+k
(c)k
, a∈R; c∈R\Z−
Using the function φp(a, c;z),we consider a function φ†p(a, c;z) defined by
φp(a, c;z)∗φ†p(a, c;z) = z
p
(1−z)λ+p, z ∈E,
where λ >−p.This function yields the following family of linear operators
Ipλ(a, c)f(z) =φ†p(a, c;z)∗f(z), z∈E, (1.7) wherea, c∈R\Z−
0. For a functionf(z)∈A(p), given by (1.1), it follows from (1.7) that for λ >−p and a, c∈R\Z−
0
Ipλ(a, c)f(z) = zp+ ∞
X
k=1
(a)k (λ+p)
(c)k (1)k
ap+kzp+k (1.8)
= zP2F1(c, λ+p;a;z)∗f(z), z∈E. From equation (1.8) we deduce that
z(Ipλ(a, c)f(z))´= (λ+p)Ipλ+1(a, c)f(z)−λIpλ(a, c)f(z), (1.9)
z(Ipλ(a+ 1, c)f(z))´=aIpλ(a, c)f(z)−(a−p)Ipλ(a+ 1, c)f(z). (1.10) We also note that
I0
p(p+ 1,1)f(z) =p z R
0
f(t)
t dt, Ip0(p,1)f(z) =Ip1(p+ 1,1)f(z) =f(z), Ip1(p,1)f(z) = zfp′(z), Ip2(p,1)f(z) = 2zf′(pz()+p+1)z2f′′(z),
Ip2(p+ 1,1)f(z) =
f(z)+zf′(z)
p+1 , I
n
p(a, a)f(z) =Dn+p−1f(z), n∈N, n >−p,
where Dn+p−1is the Ruscheweyh derivative of (n+p−1)thorder, see [5]. The operator Iλ
p(a, c) (λ > −p, a, c ∈ R\Z−0) was recently introduced by Cho et.al [1], who investigated (among other things) some inclusion relationships and argument properties of various subclasses of multivalent functions in A(p), which were defined by means of the operator Iλ
p(a, c).
Forλ=c= 1 and a=n+p,Cho-kown-Srivastava operator Iλ
p(a, c) yields Ip1(n+p,1) =In,p(n >−p),
Using the operatorIλ
p(a, c) we now define new subclasses of A(p) as follows:
Definition 1.1.
S∗(a, c, λ, p, α, β) =
(
f(z)∈A(p) :Ipλ(a, c)f(z)∈S∗(p, α, β), z(Iλ
p(a, c)f(z))′ Iλ
p(a, c)f(z)
6=β )
.
where z∈E.
It is easy to see thatS∗(p,1,0, p, α, β) =S∗(p, α, β) and S∗(p+ 1,1,1, p, α, β) =
S∗(p, α, β),is the class of strongly starlike functions of orderα and type β as given by (1.4).
Definition 1.2.
C(a, c, λ, p, α, β) =
(
f(z)∈A(p) :Ipλ(a, c)f(z)∈C(p, α, β),1 +z(I
λ
p(a, c)f(z))′′
(Iλ
p(a, c)f(z))′
6=β )
.
where z∈E.
Obviously, f(z) ∈ C(a, c, λ, p, α, β) if and only if zfp′(z) ∈ S∗(a, c, λ, p, α, β),
C(p,1,0, p, α, β) = C(p, α, β) and C(p + 1,1,1, p, α, β) = C(p, α, β) is the class of strongly convex functions of order α and type β as given by (1.5).
2. Preliminaries and Main results
In order to prove our main results we shall need the following result.
Lemma 2.1.[9] Let a function h(z) = 1 +c1z+c2z2+...be analytic in E and
h(z)6= 0, z∈E.If there exists a point z0 ∈E such that |argh(z)|< π
2α (|z|<|z0|) and |argh(z0)|=
π
2α (0≤α <1), (2.1) then we have,
z0h′(z0)
h(z0)
=ikα, (2.2)
where
k≥ 1 2(a+
1
a) when argh(z0) = π
2α, (2.3)
k≤ −1 2((a+
1
a) when argh(z0) = π
2α, (2.4)
and
(h(z0))
1
Theorem 2.2. S∗(a, c, λ+1, p, α, β)⊂S∗(a, c, λ, p, α, β)⊂S∗(a+1, c, λ, p, α, β). Proof. Set
z(Iλ
p(a, c)f(z))′ Iλ
p(a, c)f(z)
=β+ (p−β)h(z). (2.5) Thenh(z) is analytic inE withh(0) = 1 andh(z)6= 0, z ∈E.Applying the identity (1.9) in (2.1) and differentiating the resulting equation with respect to z we have
z(Iλ+1(a, c)f(z))′
Iλ+1(a, c)f(z) −β = (p−β)h(z) +
(p−β)zh′(z) (λ+β) + (p−β)h(z).
Suppose there exists a pointz0∈Esuch that the conditions (2.1) to (2.4) of Lemma 2.1 are satisfied. Thus, if
argh(z0) = −π
2 α, z0 ∈E, then
z(Iλ+1(a, c)f(z))′
Iλ+1(a, c)f(z) −β = (p−β)h(z0)
1 +
z0h′(z0)
h(z0)
(λ+β) + (p−β)h(z0)
= (p−β)aαe−πα2
1 + ikα
(λ+β) + (p−β)aαe−πα 2
.
This implies that arg
z(Iλ+1(a, c)f(z))′
Iλ+1(a, c)f(z) −β
= −πα 2 + arg
1 + ikα
(λ+β) + (p−β)aαe−πα2
= −πα 2
+ tan−1
"
kα((λ+β) + (p−β)aαcos πα
2
(λ+β)2+ 2(λ+β)(p−β)aαcos πα
2
+ (p−β)2a2α−kα(p−β)aαsin πα
2
#
.
This gives that
arg
z(Iλ+1(a, c)f(z))′
Iλ+1(a, c)f(z) −β
≤ −πα 2 , since
k≤ −1 2 (a+
1
a)≤ −1 andz0 ∈E,
this contradicts the condition f(z)∈S∗(a, c, λ, p, α, β). On the other hand if we set
argh(z0) =
πα
then it can similarly be shown that arg
z(Iλ+1(a, c)f(z))′
Iλ+1(a, c)f(z) −β
≥ πα 2 , since
k≥ 1 2(a+
1
a) and z0 ∈E,
which again contradicts the hypothesis thatf(z)∈S∗(a, c, λ, p, α, β).
Thus the function defined by (2.5) has to satisfy the following inequality: argh(z)≤ πα
2 , z∈E. which implies that
arg
z(Iλ+1(a, c)f(z))′
Iλ+1(a, c)f(z) −β
< πα
2 , z ∈E.
The proof of part (ii) lies on the similar lines. This completes the proof of Theorem 2.2.
Theorem 2.3.C(a, c, λ+ 1, p, α, β)⊂C(a, c, λ, p, α, β)⊂C(a+ 1, c, λ, p, α, β) Proof. To prove this inclusion relationship, we observe from Theorem 2.2 that
f(z) ∈ C(a, c, λ+ 1, p, α, β)⇔ zf ′(z)
p ∈S
∗(a, c, λ+ 1, p, α, β) ⇒ zf
′(z)
p ∈S
∗(a, c, λ, p, α, β)⇔f(z)∈C(a, c, p, α, β).
The proof of second part lies on similar lines. This completes the proof of Theorem 2.3.
For a functionf(z)∈A(p) the integral operator,Fδ,p:A(p)−→A(p) is defined
by
Fδ,p(f)(z) =
δ+p zp
z Z
0
tδ−1f(t)dt= zp+ ∞
X
k=1
δ+p δ+p+kz
p+k !
∗f(z) (2.6) = zp 2F1(1, δ+p, δ+p+ 1;z)∗f(z), z∈E, see [2].
It follows from (2.6) that
z(Ipλ(a, c)Fδ,p(f)(z) ′
We have the following result.
Theorem 2.4. Let δ >−β and 0≤β < p.If f(z)∈ S∗(a, c, λ, p, α, β) with
z(Iλ
p(a, c)Fδ,p(f)(z))′ Iλ
p(a, c)Fδ,p(f)(z)
6=β, for allz∈E,
then we have
Fδ,p(f)(z)∈S∗(a, c, λ, p, α, β).
Proof. We begin by setting
z( Iλ
p(a, c)Fδ,p(f)(z))
′
Iλ
p(a, c)Fδ,p(f)(z)
=β+ (p−β)h(z), z∈E. (2.8) Then h(z) is analytic in E with h(0) = 1.Using (2.7) and (2.8), we find that
(δ+p) (I
λ
p(a, c)f(z) Iλ
p(a, c)Fδ,p(f)(z)
= (δ+p) + (p−β)h(z). (2.9) Differentiation both sides of (2.9) logarithmically, we obtain
z( Iλ
p(a, c)f(z))′ Iλ
p(a, c)f(z)
−β = (p−β)h(z) + (p−β)zh ′(z) (δ+β) + (p−β)h(z). Suppose now there exists a point z0∈E such that
|argh(z)|< π
2α (|z|<|z0|) and |argh(z)|=
π
2α. Then by Lemma 2.1, we can write that
z0h′(z0)
h(z0)
=ikα and (h(z0))α1 =±iα (α >0).
If
argh(z0) =
π
2α, z0 ∈E, then
z(Iλ
p(a, c)f(z0))′
Iλ
p(a, c)f(z0)
−β = (p−β)h(z0)
1 +
zh′(z0)
h(z0)
(δ+β) + (p−β)h(z0)
= (p−β)aαeiπα2
1 + ikα
(δ+β) + (p−β)aαeiπα2
This shows that arg z(I
λ
p(a, c)f(z0))′
Iλ
p(a, c)f(z0) −β
!
= πα 2 + arg
1 + ikα
(δ+β) + (p−β)aαeiπα 2
= πα 2
+ tan−1
(
kα(δ+β+ (p−β))aαcos πα
2
(δ+β)2+ 2(δ+β)(p−β)aαcos πα
2
+ (p−β)2a2α+kα(p−β)aαsin πα
2 ) ≥ πα 2 , since
k≥ 1 2(a+
1
a)≥1 and z0 ∈E,
which contradicts the assumption that f(z)∈S∗(a, c, λ, p, α, β).
Similarly in the case when
argh(z0) =−
π
2α, z0 ∈E, we can prove that
arg z(I
λ
p(a, c)f(z0)) ′
Iλ
p(a, c)f(z0) −β
!
≤ −πα 2 , since
k≤ −1 2(a+
1
a)≤ −1 andz0 ∈E,
contradicting once again the condition thatf(z)∈S∗(a, c, λ, p, α, β). We thus conclude that h(z) must satisfy the following inequality
|argh(z)|< π
2α, z∈E. This shows that
argz(I
λ
p(a, c)Fδ,p(f)(z))′ Iλ
p(a, c)Fδ,p(f)(z)
−β < πα
2 , z∈E, and hence the proof is complete.
Theorem 2.5. Let δ >−β and 0≤β < p.If f(z)∈C(a, c, λ, p, α, β) and
1 +z(I
λ
p(a, c)Fδ,p(f)(z))′′
(Iλ
p(a, c)Fδ,p(f)(z))′
then we have
Fδ,p(f)(z)∈C(a, c, λ, p, α, β).
Proof. Since
f(z) ∈ C(a, c, λ, p, α, β)⇔ zf ′(z)
p ∈S
∗(a, c, λ, p, α, β) ⇒ Fδ,p(f)(z)
p ∈S
∗(a, c, λ, p, α, β)⇔ z(Fδ,p(f)(z))′
p ∈S
∗(a, c, λ, p, α, β) ⇔ Fδ,p(f)(z)∈C(a, c, λ, p, α, β).
Acknowledgements. We would like to thank Honourable Director Finance Sarwar Khan for providing us excellent research facilities.
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Ali Muhammad
Depatment of Basic Sciences
University of Engineering and Technology Technology Peshawar Pakistan